On a Cheeger type inequality in Cayley graphs of finite groups

Abstract

Let G be a finite group. It was remarked in [BGGT15] that if the Cayley graph C(G, S) is an expander graph and is non-bipartite then the spectrum of the adjacency operator T is bounded away from −1. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval −1denotes the (vertex) Cheeger constant of the d regular graph C(G, S) with respect to a symmetric set S of generators and γ = 2 9 d 6 (d + 1) 2 .This is also called the vertex Cheeger constant of a graph.

On the Bipartiteness Constant and Expansion of Cayley Graphs

“…Now by combining our results with Trevisan's bound, we obtain the following theorem that improves on the main result of [3].…”

“…For this step, we will use a technique developed in [9], which was used to prove similar results in [3,4,6]. Without loss of generality, assume that |L| ≥ |R|, and notice that |L| ≥ 1−ε−βout 2 n. Suppose that |L| ≥ 1+ε 2 .…”

“…Combining this result with the Cheeger inequality, it is seen that max i>1 |λ i | is bounded away from 1. Biswas [3], building on that argument, gave a bound of the form 1 + λ n ≥ h 4 out /[2 9 (d + 1) 2 d 6 ], and in very recent work Biswas and Saha [4] refined the bound to 1 + λ n ≥ h 4 out /(2 9 d 8 ). In this paper we improve on these results (see Theorem 2.6 below), by proving that for every non-bipartite Cayley graph, 1 + λ n ≥ Ch 2 out /d 2 , where C > 0 is a universal constant.…”

“…A brief outline of the overall proof strategies is as follows. The proofs of the previous results [3,6] involve, for the Cayley graph G = (X, S), examining the multigraph G 2 = (X, S 2 ) with edges consisting of 2-walks in G. Then letting A be the h out (G 2 )-achieving set, the outer (vertex) boundary…”

A Cheeger type inequality in finite Cayley sum graphs

“…One has the following corollary of Theorem 1.4. As a by-product of our proof, we improve the bound established for Cayley graphs in [3,Theorem 1.4]. See Theorem 2.12.…”